H. Hanada, J. Kado, and K. Oshiro have introduced, in a diagram of modules and homomorphisms, the concept of generalized <i>M</i>-injective modules. S.Mohamed, and B. Mueller have given a different characterization, based on an exchange property, of the generalized M-injective modules. Here we introduce the concept of <i>M</i>-jective modules, which is a generalization of Mohamed and Mueller concept for the generalized M-injectivity. The concept of <i>M</i>-jective modules is used here to solve the problem of finding anecessary and sufficient condition for a direct sum of extending modules tobe extending. In fact, we show that relative jectivity is necessary and sufficient for a direct sum of two extending modules to be extending. Wealso introduced the concept of generalized extending modules, and give some properties of such modules in analougy with the known properties for extending modules.
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Acta Mathematica Universitatis Comenianae, Institute of Applied Mathematics (provided by DOAJ)
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